CISC/CMPE 223 - Assignment 4 (Winter 2022) Due: Thursday March 24, 2:00 PM Regulations on assignments · The assignments are graded according to the correctness, preciseness and legibility of the solutions. All handwritten parts, including figures, should be clear and legible. This assignment is marked out of 20 possible marks. . Please submit your solution in onQ before the due time. The submission must be in one of formats: . PDF, .JPG, .PNG, .DOCX. · The assignment must be based on individual work. Copying solutions from other students is a violation of academic integrity. See the course onQ site for more information. 1. (5 marks) What should the pre-condition P be in each of the following ten correctness statements for the statement to be an instance of Hoare's axiom scheme? All variables are of type int. (a) P { x = 1; } x == 1 (b) P { x=1; } x == 2 (c) P { x = y + z; } 0 < x + y + Z (d) P { x = y*z + 3; } x*x > y + 2 (e) P { z = y + 1; } Exists (z = 0; z < 10) z + y == 50 (f) P { x =x+y; } Exists(y = 0; y < 15) x*x == y + t (g) P { x = y + 1; } ForAll(z = 1; z < 100) x + 2*y > z + 2 (h) P { z = x + y; } ForAll(y = 1; y < x) x + y + z < 100 (i) P { x = y + z; } Exists (x = 0; x < 10) z*z + 2*x == 15 (j) P { x = y + z } Exists (y = 0; y < 100) (x + y == 15 | | z*x + y < 100 ) 2. Verify the validity of the following correctness statements by adding all the interme- diate assertions (that is, give the proof tableau). All variables are of type int. Clearly state any mathematical facts and inference rules used. (a) (2.5 marks)
ASSERT( x >= 5 || ( x <= 0 && y == 2 ) ) Z = x - y; y = y + z; X = y - z; ASSERT ( x == 2 | | y > 3 ) (b) (2.5 marks) ASSERT( z >= 0 ) if ( x < 2 ) { y = z+3; } //end-if Z = x + Z; ASSERT( y > x + 1 || z > 1 ) 3. (5 marks) Select a loop invariant and give a complete proof tableau (that is, add all the intermediate assertions) for the following correctness statement. Also make an argument for termination. All variables are of type integer. ASSERT (k >= 0) i = k; sum = k; while( i > 0 ) { i = i - 1; sum = sum + i; } ASSERT (sum ==